Functional limit theorems for the maxima of perturbed random walk and divergent perpetuities in the M 1-topology
Tóm tắt
Let (ξ
1, η
1), (ξ
2, η
2),… be a sequence of i.i.d. two-dimensional random vectors. In the earlier article Iksanov and Pilipenko (2014) weak convergence in the J
1-topology on the Skorokhod space of
$n^{-1/2}\underset {0\leq k\leq [n\cdot ]}{\max }\,(\xi _{1}+\ldots +\xi _{k}+\eta _{k+1})$
was proved under the assumption that contributions of
$\underset {0\leq k\leq n}{\max }\,(\xi _{1}+\ldots +\xi _{k})$
and
$\underset {1\leq k\leq n}{\max }\,\eta _{k}$
to the limit are comparable and that n
−1/2(ξ
1+… + ξ
[n⋅]) is attracted to a Brownian motion. In the present paper, we continue this line of research and investigate a more complicated situation when ξ
1+… + ξ
[n⋅], properly normalized without centering, is attracted to a centered stable Lévy process, a process with jumps. As a consequence, weak convergence normally holds in the M
1-topology. We also provide sufficient conditions for the J
1-convergence. For completeness, less interesting situations are discussed when one of the sequences
$\underset {0\leq k\leq n}{\max }\,(\xi _{1}+\ldots +\xi _{k})$
and
$\underset {1\leq k\leq n}{\max }\,\eta _{k}$
dominates the other. An application of our main results to divergent perpetuities with positive entries is given.
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